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# Lec15 - I E/Stat 265 Fall 2009 I E/Stat 265 Fall 2009...

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Unformatted text preview: I E/Stat 265, Fall 2009 I E/Stat 265, Fall 2009 O /Stat 65, a 009 O /Stat 65, a 009 Lecture #15: Lecture #15: Estimation Concepts and Methods Estimation Concepts and Methods- Concepts of Point Estimation s a o C o p s a d o d s s a o C o p s a d o d s 6 1 Concepts of Point Estimation A. Biased Vs. Unbiased Estimators B. Minimum Variance Estimators C. Standard Error of a Point Estimate 6-2 Point Estimation Methods A. Method of Moments (MOM) 1 B. Method of Maximum Likelihood (MLE) C. Method of Bayes (Not testable this term) 1 Point Estimates 6.1 Point Estimates ¡ Objective – obtain an estimate of a population parameter from a sample (e.g. sample mean, , is point estimate of population mean μ x ) x ¡ Applications: ¡ Parameter Estimation ¡ Hypothesis Testing (and other inferential statistics methods) 2 arameter Estimation Example Parameter Estimation Example uppose you wish to estimate the population ¡ Suppose you wish to estimate the population mean, μ ,. Some possible estimators include: ¡ Sample Mean, Sample Median, Sample Trimmed Mean, Sample Geometric Mean, Sample Harmonic Mean, (Max X + Min X)/2 ¡ Recall example from descriptive statistics, which of the following is the “best estimator”? ample Mean = 965 0 Sample Median = 1009 5 ¡ Sample Mean = 965.0 Sample Median = 1009.5 Sample Trim Mean = 971.4 3 arameter Estimates it Parameter Estimates - Fit ˆ ¡ Estimates, , always have measurement error. hy? estimation error ˆ θ = θ + θ ¡ Why? ¡ Because is a function of observed x i ’s (r.v.) that change with n ˆ θ ¡ We identify the “best estimator” as the one with: inimum Bias (unbiased) vg stimation error ¡ Minimum Bias (unbiased) → avg estimation error = 0 ¡ Minimum Variance of estimation error → 0 as n → ∞ ¡ More consistent predictions 4 p ¡ Increases the likelihood the parameter estimate represents the true parameter iased Vs Unbiased Estimator Biased Vs. Unbiased Estimator ¡ Bias - difference between the expected value of the statistic and the true parameter θ ˆ θ nbiased Estimator: ˆ Bias = E( ) θ − θ ¡ Unbiased Estimator: ¡ Example: is an unbiased estimated of μ (Bias = 0) ¡ Suppose X 1 , X 2 , .. X n are iid rv’s, with E(x i ) = μ x n i i 1 E(x ) n ˆ E(x) E( ) = μ = θ = = = μ ∑ 9 n n nbiased Estimator of Variance Unbiased Estimator of Variance hich of the following formulas is an unbiased ¡ Which of the following formulas is an unbiased estimator of variance, σ 2 ? n n 2 2 i i 2 2 i 1 i 1 1 2 (X X) (X X) ˆ ˆ s ˆ n 1 n = = − − θ = = θ = σ = − ∑ ∑ ( ) ( ) ⎛ ⎞ − = σ σ = σ ⎜ ⎟ ⎝ ⎠ 2 2 2 2 n 1 E s E ˆ n ¡ What happens to the bias effect as n becomes large?...
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Lec15 - I E/Stat 265 Fall 2009 I E/Stat 265 Fall 2009...

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